CHAPTER II . I N TERACTION OF RADIATION WITH MATTER k. ENERGY LOSS BY CHARGED PARTICLES 1, A charged particle moving through matter loses energy by electromagnetic interactions which raise electrons of the matter to excited energy states. If an excited level is in the continuum of states the electron is Ionised; if not, the electron Is in an excited bound state. In either case the Increment of energy is taken from the kinetic energy of the incident particle. In the following section "ionization" will refer to both degrees of excitation. Range - total distance traveled by the particle until its kinetic energy is 0. Before a formula for the range of a par- ticle can be derived, the rate of energy loss per unit path must be calculated. The first such calculation. Is due to Bohr*, and is essentially classical, I.e., non-quantum mechanical. 2. Bohr Formula. Consider one electron of mass m at a distance b from the path of an incident particle having charge Je, mass M and velocity V. a (electron) — fo= Impact parameter mass M charge £e»- velocity V A FIG-. II. 1 Assume the electron ts free and initially at rest, and moves so slightly during the collision that the electric field acting on the electron due to the particle can be calculated at the initial location of the electron. The last assumption Is not valid for an Incident particle of velocity comparable to that acquired by the electron. We shall calculate first the momentum acquired by an elec- tron during a collision, and from this find the energy acquired. As the particle passes, the electrostatic force F changes direction. By symmetry the Impulse Jf (1 dLt parallel to the path is zero, since for each position of the particle to the left of A, yielding a forward contribution to the impulse, there Is a position at equal distance to the right of A giving an equal but opposite contribution. t-J F, dt We first estimate ,*e' b The impulse -L to the path is the order of magnitude of %± '■ If. = J Fi at ^(electrostatic force )X(time of collision)--^- Ky" More exact computation: Consider a circular" cylinder centered on the path and passing through the position of the electron, Fig. II. 2, Let S be the electrostatic field in- tensity due to the particle VS. The electric flux Is Js-ds = 4Tr^e (independent of V) * Bohr, Phil, Mag . 24, 10 (1913)', 10, 581 (1915) 2? 2C Bohr Formula Ch. II toy Gauss's theorem. If S x = component of £ _L to the path, then the flux = f% Zv ktl = "tff^ft. Therefore f^jisf^^S , The variation of £1 with time at the electron Is the same function as would toe found by keeping J-e fixed and observing at a point moving with velocity V along the cylinder surface. Therefore T\ib T FIG. II. 2 The Ins pulse Ii."i«, £j.€«-*~ yV = *'■■ momentum acquired by the electron, electron is then 2_nrL omV 2 b z where p is ike II . ? The energy acquired by one II The number of collisions per unit path length such that b lies in the range to to b + db is equal to the number of electrons per cm, length in the shell bounded by cylinders of radii to and b + db (Fig. I 1.2). If % = number of electrons per cm3, this Is 2n "ft bdb. The energy lost per cm. to such electrons is cLE(b) - — ^4-7rfe + ^ ^b rm V' II. 4 verge ;m ) II. 5 The total change in energy to all shells in the range b min t0 b max lB The limit b max : The force on an electron as a function of time will be a pulse occupying a %±mp *%■=■ b/V , non-relativistically , It can be shown that if l/£ is much less than the vibration fre- quency 2/ of an electron In an atom, then the electron absorbs no energy, i.e., the probability for transition to a higher state Is small , Relativistic ally, the electric field of the incident particle Is contracted in the direction of motion and S± is increased by a factor Vf\ -/3'- A~% This "sharpens" the impulse given the electron; the duration 'V of the pulse of force Is now approximatelyY^^ifTrgF . The Integral of the Impulse is not changed, since it depends essentially on the product (field strength)X(duration) . Ch. II b T Bohr Formula 1/ ^— _. b\|7^3= 29 Tjg^ is chosen so that /^ >7^ range integrated. Thus we may set b_ = V V (cm) is valid over the II. 6 where W is an appropriate average freauency for electrons In the absorbing material. Problem: Discuss the statement that for \/ v <T, energy transfer to electrons Is negligible (or indole of adiabatic invariance) . Consider the component of the motion of the electron J- to the path of the incident particle. Let the coordinate of the electron be y. y = b + d sin^t. y = jy d cos V t. b»d. A (energy) = {j component of force )X( velocity ) dt Since cos 6^1 r Cfb) © A( energy) < f^P^j^J^pq^dt = Vi T I 1 Y 2 -i,M &*+/<'' V a as 7^r->»i The limit b min : (l) Classically, the maximum velocity that can he imparted to the electron (in head-on collision) is less **~ than 2V.* The energy given cannot exceed 4m(2V) 2 . Therefore b cannot have values that imply a greater energy transfer per collision than 2mV- As a function of b, the energy transferred collision la |^ . Values of b smaller than the solution coll oer This determines b mln as must be excluded in the integration, ffotfl J£ [1.7 ' M i. (2 l + ^ hl f claGSlcal - treatment is valid only if the Coulomb field oi the incident particle varies negligibly over the atraeii- 3i on ^ ot the quantum mechanical wave packet representing the * H 11 ? ls ea3 ie 8 "t to see in the rest system of the Incident par- ticle. Then the electron apoears to collide with something like a rigid wall. 30 Bohr Formula Gi- ll electron, k Is approximately the do Broglic wavelength of the electron as seen from the Incident particle. In a coordinate system In which the incident particle is at rest {this nearly coincides with the center of mass system , for a heavy Incident particle), the electron has velocity of about V, aBSUffilng its orbital velocity is much less than V. The momentum of the electron in this coordinate system is mtfj/sfiZg* and therefore &** ~^W • Only values of V>4\ have meaning, and therefore another criterion for b mm is k M3I ritiV II. 8 The larger of (b Eln ) d and (b mln ) aM should be used in the integration*. For values of V and therefore II. 8 whould be used ere o max -> Using II ,.; •in* ( b min)> ( b minL i in II. 5: *" II. 9 where V is a suitable average of the oscillation frequencies of the electrons, .More precise calculation** leads to the following formula for heavy particles, I.e., not electrons: r (erg cm ) (heavv particles) 11.10 where I is the average ionization potential of the electrons of the absorber, in ergs. The In term as 9 for 1 mev protons in NTP air . 3. Electrons . There are two main reasons why 11.10 canr.ot apply to electrons. (l) The derivation assumes that the Incident particle Is practically undeflected. But the Incident particle acquires a transverse component of momentum per collision approx- imately equal to that given to an electron In the absorber, and if the incident particle Is an electron, the transverse velocity corresponding to" this momentum will not be negligible. (2) For collisions between Identical oarticles exchange phenomena must be taken into account*--*. Eethe *'# gives the following formula for energy loss by electrons: oLx/ lw . V^ t 2I 2 073 a ) s / } ___^ . (electrons) 11,11 where I is the average Ionization potential of the atoms of the absorber and T = relativistic kinetic energy of the electron. *Ir. cutting the Integral of II. 4 off at \ ln ^O, we have neglected a term / bm ' T1 - cL E(b) . This is lust If led In "Lecture Series in Nuclear Physics", LA 24, Lecture XI, printed edition p. 2?. ** Bethe, Handbuch der Fhysik , p. 519 ***Mott, Proc. Rov . Soc. 125 , 222, JK26 , 259 (1929) Ch. II Bohr Formula, Electron: Ari approximation for I ig-J? (■ergs) 11.12 val U ef; iv f ?fS%g-g b 2! Sl ^*™> ****** *> air, ^ the ** -Ot her particles. For incident particles of identical charge moving in like absorbers -a_E/dx is a function of V bhlv Therefore if -dE/dx as a func- ' t-ion of enernv is Vatintfi for, say protons of mass y it can be tound for some other singly charged particle B by changing the energy scale so that the new Energy, in e . y . leu 10° 10 I 108 10 ^ 10 10 Electrons in Ai r ., dE in e.v = per ^X__^gZcm^ 6 19.5 x 3.67 1.69 1.95 2.47 2.79 3.48 LQ energy values are K B /M D times the old. The following table Cl) de r uS?o 3 r.f TXV n 4w enatoleS ^ lllnS ° Ut thG table f °^ energy correction is ™a p v,,+ J i+\, lerent mass, the above valuers multiplied brrMf-^si "^ IT™^ l0SS * Uk/*/- > B faince fy enters the formula. ergs per ,ram cm^^ S^oted^ Gr l0GB ln thiS table is where f is the thickness in a- CB r a . Fo- thi* 1S n °i : 'o ain3d ? r ?S ^ E ' 7df b ^ '"ultiolvin e rg~ c n: of air, 11.13 tab! e -dE/dx i n g by the density PROTONS" In Air Energy , _dE Mgy d^ -cm" 300 47 7.6 2.3 ^^^PP^X^^OlSi^^ e. v. 2xl0 6 JL cv jT e.v. .... 300 2x101 47 2xl0« 7.6 2.3 2.3 2x10- 2xlD 10 - ,- -■ -- - — £U£LJ^L ,117X10° 300 ,4x10^ 1200T106 iftQ .117x107 47 4xloI ' l "~"'- '■"' .117x108 7.6 4xl0 8 30 .117x10- 2.3 UxlO 9 c 2 .UTxlO 10 g,3|4 xl0 10 I 2 188 lllOl 27,^ 10 a 5 10 9 ] .6 ilO 10^1.6 portionaf^c^e^dfn-sTfv ot^fS^^^ l0SS W " , Xact ^ P™~ £" c ^ 2 ^r a given "articK ^ d SJ**** ^2 1^ e ^ POT on two further fee tors- m K Val 2' But " dE /^ denends ? of an atom does no^ increaS T S^' e ^trons rer atom; in the formula for (ffi/dS J?l e 1, ?i ? ' ? 'f ths v * 1 £S ht -- thusj?^ appearing in the in - t=t > f heavier elements. (2} I C the -°'- term ' depends on the absorber (Eqn. 11.12)4*. Wx&rJt^ tSfs e ^« n . e f£ erl f eilts the original, data is the ... na iron, tni. the energy is estimated. We have derived an See also Livingston and Sethe , Bev . Kod . Phys . 9, p. 26 5 . m Bohr Formula, Range Ch. II equation of tne for, -dE/dx = f{l). Integrating, we S et II .14 X as the range of a particle wxwi * rough approximation nation ^y teP^formM numerically Fpr a ^ ug_ no ^ elatlvl ti _ we assume f(E ) °Cl/V^l/E. Ihen k • o ' bet ter approxiraa- cally. More precise consideration showB taai a tion Is* % _ w3 U* Ef^V 11.15 jk___,o ».*.< A rourli formula Giving Empirical range- energy formulas . A roij ^ atmog pheric range of alpha particles in air at 15 the oressure is 36 ■ft*s.32 (Mev) £ C>vl. ( alph aB in air) II. 16 IMa is correct to ahout 10% ™^$J^J^™%™£*te velocities. The general nature of tne range enei^y shown in Fig. IT»3- 7 Poi-rization_^ffects^ In the T< derivaUon^f-IlTlO no ^^^.^f taken of the Influence on one electron due to the simultaneous motion of tne otner electrons near it The elect rons in a region move so as to diminish the electric field beyond that region This partial shielding effect increases with increase in density of *l*ot*onB. ™> change in -dE/dx due to tnls effect Is usually small.** If the index of refraction, n, Is net one.'the velocity of light is less than c example, n~1.5 and velocity^ light^ is^ J/2 FIG. II -3 In water, for c . If the In- a, particle produ. - its experimental discoverer*** 8. Nature ^f^e_eouation for -dE/dx. Equation II .10 has the form ^T^ne^or^^ Livingston and ^,e B afeS e in P defali hy Fermi ^ -,%£&* &$&, l.C ***Cerenkov , Ehvs. Rev. 3E« 378 lUMJ ' o. 261. [I Ch. II Ionization of a Gas 35 W) 30ANTITAT1VE CURVES IN MDNI'GOME RY, COSMIC RAY PH VSICS } p 35 a 10 Mc 1 FIG. II. 4 lOOMc 2 ENERGY The curve BCD gives the l/V dependence. At relativistlc energies V changes little and CD Is asymptotic to V = c. At relativistic energies, the log term in (V-/l-f5 2 ) changes, and increases as V— >e, giving the rise in the curve from to E. At very 1 ov,' energies (region AB) equation 11.10 "breaks down because the particle has "velocity comparable to that of the orbital electrons in the absorber, and the efficiency of energy exchange Is much lower. The particle itself captures electrons and spends part of Its time with reduced charge. 9. Ionization of a Kas . If ionization is produced in a gas the ions may be collected by charged electrodes, and the amount of charge collected will be proportional to the number of ions produced. The change of potential of one of the electrodes will depend on the charge collected (and the external circuit) and therefore on the number of Ions produced. This voltage pulse may be amplified linearly and measured quantitatively, a 3 with an oscillograph. A gas chamber for this purpose Is called an Ion- ization chamber - " . + HIGH VOLTAGE R ' LINEAR AMPLIFHES ZL OSCILLOGRAPH FIG-. II. 5 In the arrangement In Fig. II. 5, electrons are collected at the top plate. A 'negative pulse,. of duration determined by R and the capacity of the ionization chamber and associated circuit, is produced at the grid of the linear amplifier. It turns out that there Is a close proportionality between number of Ions produced and total energy lost by the incident particle. For most gases one Ion pair (electron plus ionized atom) is produced for each 32- 3^ e - v - l03t b y the particle, (see table or following page) . Although empirically the result Is a * Reference:; on ionization chambers are: Korff , Electron and Nuclear Counters (Van Nostrand), Rossi and staub , Ionizat ion Chambers and Counters (McGraw-Hill) . 34 Scattering Ch. II simple proportionality between number of ions and energy spent , the explanation is very compli- cated. Theoretical prediction of the average energy per ion pair involves: (l) calculating the % of all primary collisions that lead to removal of an elec- tron in order to know how much energy is "wasted" on nor.- ionizing excitation of the atom; (2) cal- culating what fraction of energy carried away by primary ionized electrons is used secondary ionization. This problem has not been investigated , En e rgy for one ion pair* Gas Energy spent for one ion palr,e.v H 33.0 He 27.8 N 35.0 32.3 Kc 27. 4 A 25,4 Kr 22.8 Xe 20.S in producing completely Problem: Design an ionization chamber and snecify the char- acteristics of an associated linear amplifier so that the system is suitable for measuring the a energy difference ■"■' (UI) and U 2 34 fijll) a d< between the U decays 92 1 a + 90 Th 234. 90 qiV, -„234 -2S~+ 92 234, 92 -^34 tr- .a +9Q Th 230 The following are among the necessary considerations: £1) loss of energy of particles while still in the emitting substance, (2) gas to be used, (3) dimensions and electrostatic capacity of chamber, (4) gain and frequency response necessarv In the amplifier, (5) rate of emission of partiales by the emitter. 10. High energy g parti cles lose energy mainly by radiation. This effect is taken up later, in II, section C, 3. B. SCATTERING DUE TO A COULOMB FIELD Scattering due to interation of charged oarticles with the Coulomb field of nuclei is distinguished from* scattering In which the incident particle enters a nucleus. Only the former is treated in this section. Scattering due to c .all charged particles In cloud chamber may have a angle scattering event, quently, and their track /3 — 1, Classical calcu] ollisions with nuclei is observed for varying degree. An alpha track in a single kink, indicating one large Electrons are scattered much more fre- s are as shown: o< ~N screening of the nuclear the force on an incident Coulomb force ^%&yOl % and J- the charge of the heavy compared to the oa system is almost at the and , the angle of defl ation for single scattering. If the charge by nearby electrons is neglected, particle due to one nucleus is the , where g is the charge of the nucleus particle. Assume that the nucleus is rticle so that the center of mass of the nucleus. Let b, the impact parameter, ection be defined in Fig. II. 6, p. 35. For inverse square forces between particles, classical ^Rutherford, Chadwick and Ellis, Radiations from R adioactive Substances , p . 8 1 . ' Ch. II CHARGE ft' Scattering 35 NUCI-EUS OF CHARGE 2 MASS M VELCC 1TY V FIG. II. 6 mechanics gives the following formula* TaN 2=^W* 11.17 This formula is valid at non-relativistic velocities, V<< c. A relativist ioally correct version of 11.19 for small angles 9 is given in the paragraph containing 11.20. Exact quantum mechanical calculation gives the same formula provided the nuclear field is exactly a Coulomb field. Both classically and quantum mechanically the formula is valid only If the distance of nearest approach of the particle to the nucleus Is larger than the nuclear radius . The cross- section for scattering of the at an angle 9 in the range 19 is defined to _L to the Initial path of the particle such passes through this area It Is deflected tay Since for given particle and nucleus, b is a the area corresponding to a given & lies at a and has magnitude dC% = 2tt bee) db. Substltu of the corresponding angle 8: do^ = 27tb(0)b the element of solid angle 27T3ln9d9 to find unit steradian, and substitute for b(e) Its 11.17 for b. Then the cross- section per uni is incident particle be the total area that If the particle an angle 9 in d9 . function of only-. certain radius b{9), tins for c In terms 1 (ej&fi . Divide by the. cross section per value from solving t solid angle at dcr cLcu _. i (e a H 3f I 4\M V 2 / Si* &% 11.18 Note that most particles are scattered at small angles. A relativi stically correct equation for 9 as a function of b when 9 is small can be derived easily, using the same argu- ments used to derive II. 3. Since now v:e deal with nuclear charge 2 and incident particle of charge ^-* , we must multipy II. 2 by the nuclear charge In order to get the transverse Impulse Impart- ed to the incident particle in the collision. This gives AJo=^£^ 11.19 v yb If As la the relativlstic momentum of the incident particle, angle of deflection Is very nearly Z^Jp/Jp , if &Jp«Jp. the e (e sma 11) 11.20 If we put fsMV (non- relativi stlc) this becomes identical to 11.17 when 6 Is small and tan 9/2 a^ 9/2. In these formulae *See, for example, Lindsay , .Physical Mechanics , p . 76 • ^ 36 Multiple Scattering Ch. II b is limited to distances from the nucleus within which the nu- clear charge can "be felt, i.e., has not been screened ay nearby electrons. 2 . Multiple Scattering. Particles , particularly electrons , are deflected many times in passing through a foil of metal. The net angle of deflection, denoted @ , is the result of a sta- tistical accumulation of single small scattering events. The de- tailed theory is complicated . A simplified treatment will be given here. We assume that no paths are complete loops, as in b FIG. II. 7. It is plausible and can bo shown** that the values of © for many traversals are distributed about ©-O according to the gausslan lay;, i.e., probability f or © in the For small *s ©f Lty scattering angles , e range d.® is 11.21 ® P = Z 0^ 11.22 Dp is the net angle of deflection for p collisions and the bar means the average for many such traversals of the absorber. Since statistically the individual events do not differ, ~6? =&* II. 2 j FIG-. II. 7 (gf=2TrND Using II .20_Tor 0, averaging over Q 2 for values of b from b n j_ n to bmaxj an< 3- summing for all collisions in the length of path U, we obtain ***» W )uh 11.25 of the foil. For thin foils, D differs little from the thickB.es N — no. of atovns uer cc. Due to screening of the nuclear charge by electrons, the 2 felt by a particle depends on b, therefore Z in the integral is a function of b. We take 2 outside and adjust for the error by choice of b^. Assuming the absorber is so thin that _Jp and V do not change , Choice of limits: b max : The equation 11.26 would be strictly correct if at distances beyond b^ax the screening of the nucleus were perfect; i.e., no scattering, and for distances within b m ax there were no screening at all, i.e., full value of 2 uere felt. No such boundary exists, since screening in- creases gradually with distance, but fortunately the log term * E. J. Williams, Proc . Roy . Soc . A169 531 (1939); Rossi and Greisen, Rev . Mod . Plrys . 13 249 (1941), ** Rossi and Greisen, I.e. *** This is shown in Appendix II ,1 «*■»* Shown in greater detail in Appendix 11,2 Cl1 ' T1 Multiple Scattering is not sensitive and we mav cut* l_ Bohr radius q gS V* 37 11.27 me factor ? a takes into account the variation of the function radius/f ) e4€ " Z ~ (electron charge within sphere of The limit "brain effectively adjusts the maximum angle in a single scattering process. Since we are not "counting any values of © > i ,we may impose the rough restriction that £?<1. This glims --U&HT ATOMS HEAVy ATOMS 2^e ! FIG-. II. 8 !*>« V-^o (from 11.20) 11.28 Otner considerations may govern b^ such as (l), the finite size of the nucleus: b mln >A / 3 1 . 5 x 10 ~13 j frora ePuatlOT] 1A ? G L° ;; UJ ' 3ize of _™ye packets of particles in tbe colli- f 3 ° n **- Tne result for @* using our choices for b min and. b^ Jn a Vyb 11.29 The result Is not sensitive to the choice of b mln and b max ; the log terra is of the order of magnitude of 10. JuTae to thf°|« fac- tor, scattering increases rapidly with weight of the elements in the absorber, even for equal atomic densities. n tivi^^^S^^C f, X f kinetlc energy). In the extreme rela- tivists range \jjp*^ kinet i c ene rgy . The re f o re , roughly , w KINETIC ENERGr 11.30 This formula gives the :nost important characteristics of 11.29 Evidently a proton and an electron of the same energy wil n be ff^ ed a °? Ut the . sanie amount. "But the range of the electron is ,uch greater, ana therefore, experimentally , large anrle scattering of light particles is more prominent than of heavy particles. . ' J Nuraerical formulae: ELECTRONS IN LEAD &' £*io e D (KILOe.v.) 2 11.31 ladiaT^^f' 311 S'^aS 1 MeV el « ctron ^ delated about one radian in passing thru 10^3 em. of lead. Electrons iw MK (B) 2 j 70oo D '(kilo e.v.)' 11.32 This is the radius of an atom of charge? using the Thomas Fermi model; cf. Lindsay, Physi cal Stat i st 1 c a ■ ^ . 226 tK^ ar V ls ?" s ^ in T.dlliaras, I.e., and rn^re briefly, in J&nossy, Cosmic Rays , para, 173. J ' 38 Passage of Electromagnetic Radiation, through Matter Ch.II scattering in the mind, than the thickness of t\ Lnd absorption must be interpreted with The true oath lenght mSv be much longer le absorber. The equations for range of particles applied to a homogeneous beam of non- scattered parti- cles passing through an absorber would yield a curve of number of particles emerging vs. thickness like FIG. II. 9a. But ex- perimentally, FIG. II. 9 b ^s more nearly correct for electrons, and the reason for the difference is straggling due to scatter- ing;, f NO. EMERGING f NO. IN ORIGINAL -^ RANGE - THICKNE5S FIG. II. There is a problem on multiple scattering in Appendix II. 3 C. Passag e of Electromagnetic Radiation through Matter The] ;d Oj ill There are three processes by which photons are either ab- sorbed or lose energy: Photoelectric absorption Compton effect (3) Pair formation. 1. Pho toelectric a bsorption. The electrons bound to an atom may be grouped into shells according to their binding energies. The Innermost shell, the K shell, contains the two Is electrons and is bound with an energy of about Ry (Z-l) 1 -, where Ry is the Rydberg, about 13-5 e.v. The next shells. are the L and M shells. Electrons in the L and M shells are bound With energies roughly Ry (2-5 ) 2 A and Ry (Z-13) 2 /9, respectively. These last two numbers are very approximate, since the energy de- pends on screening and the shape of the orbit. The last filled orbits of the atom are the valence orbits producing optical spectra. Beyond this, the states are unfilled. Provided its energy is sufficient, a photon may remove an electron from any of the shells and leave it either in one of the previously un- occupied bound states, or in an ionized state. As the frequency p of Incident photons is increased from zero, they first become able to excite electrons in optical orbits of a feu e.v, binding energy. This loss of e-ergy is the first contribution to absorp- tion of photons by photoelectric effect. Asj/ is further increas- ed, electrons in deeper lying orbits may absorb energy. When V reaches the binding energy' of "a particular shell of electrons, there is 'a sharp rise in absorption. A rough plot of the ab- sorption as a function of V is given in FIG. 11.10. The point ab so rot ion Lt whl eft th e a h so rp t i on change s abrupt 1 y du e to, sav, absorption by K electrons is called the K absorption edge, and is at the energy at which K electrons are lifted to energy levels > 0. Energy is the energy zero velocity. of an ionized electron with sub The L absorpti ■shells 25, 2Pk or. edges are three-fold, since the L shell has md 2Pa/ having different energies. This Oil. II Photoelectric Absorption 39 energy splitting, is due to (1) different screening of the nucleus for S ana P orbits, and (2) electron spin-orbit interaction split- ting of Py and Py . -Z- Q fc- Q- CC O u~> CO < K EDGE DECREASE -AS p> ^B.ET. f k ORBIT, FIG. 11.10 ■ 35 KILO e.v -for TIN Z=5Q Tne cross-section for photoelectric absorption by one K shell electron (for transitions to the continuum of energy >0) has been calculated by quantum mechanics, assuming a hydrog<=n- like wave function for the electron (Is) . The result is* (3 (jW) ^er tatlectvon = I28TT e' 3 e -4scot-'e m& zy+ |— q- •zve 11.33 where Z^ .. is the frequency of the K absorption edge, 1/ is the frequency of the photon and is greater than l£ , and £=\Tj^ The following simplified formula holds for V near to t^ : &fe (5 p (tV) jaer Is electron «~ S.3:' 'O'^JZJ^ z z Mtteir to 1A. 11.34 The mass absorption coefficient A is defined hy the relation (fraction of photons not absorbed when passing through thickness § m g-cm tJ of absorber )se4>k*f). A s i m ii ar equation defines the absorption coef f icient x . i.e., the fraction massing through thicxness x m cm is equal to ~-xx " fc; Problem: Derive the relation between >u and cr per atom Cal- culate^ for lead. y Consider a beam of/n, photons-am -2 - sec -1 impinging on a sec- tion of absorber of thickness dx and cross-sectional area S An area B given hy B = V ( cm^ ) X N (cm" S ) X S (cm 2 ) X dx (cm) may be considered to absorb all photons hitting it. B£s. The number of photons absorbed per second is (B/s)^ , The emerging team has density^ per cm 2 per sec' g <ffa = -dHcbx sk~sw e —trMa. A STL The fraction net absorbed in a finite distance x (cm ^ = P^ N ^ = ^^f" m = C is by definition. Heitler, quantum Theory of Radiation , 2nd ed/p. 124 ■■-■ Photoelectric Absorption cii. II But r*»y where /> is the density in g-cm-3. Therefore ^ = f(^Y)"-^ m W™ ^ N = £* ,o 23 11.35 £^ f b * °^culated by equation 11.33, multiplied by the num- ber of K electrons P er atom, i.e., £, an d^ found by II 35 Tills calculation ignores the contribution^! the L^M N etc' lit) ?h, f C r?i lnS t0 Rutherford . Chadwick & Elli . ;, 4 n ^ v S ? e ii v aCC T ° T U ? tS ** -a* of absorption for photons ene?Sy 5 19 E 3 ^ ^ S e ^i° n 11.34,^/ for lead at photon energy- .19 mc la -.,-6 en--g X, Actually,* is— 8 In 1 x nr, 1 ng^ , 5 . 76 , we i gno red ab o r ?t i o n in other than K shell. tth change "of K The sharpness of the absorption edges nohes pee methcu of measuring the frequency of ^ rays of low e J edges yary for different elements, and one may det tween which two K edges the unknown nhoton erern; 11 lng the sudden change in absorption ~ ' of the absorber used. sible ■■ =1 LiJ ■ ermine e s by- edge The e- ohse re- location a* C u S n i D l \\ * 1 \ 1 \ .Oc >:5 .0 2 .& \ 7 . 9 rNERGY IN UNITS OF "mc* LOCATION OF K ABSORPTION EDGES PIG. 11.11 Proolem: Giver, a source of 05 kilovolt H radiation. What absorbers would you use to bracket this energy and thus~ serve to determine it? Determine the thickness to be used. Qlve r™n Q8 l0n t0 thS Pliability of the chosen absorbers. Compounds may oe used. The absorption of a compound is the sum of the separate absorptions due to the constituent element s. 3 . '-&V in. Comoto n scattering . Photoelectric ab so rot ion decreases r-eases. Soon after the K edge is massed the oredomlnant process removing energy fro.-, a beam is scattering by electrons Si iV?\ true abso ^Ption, since most of the energy Is ro? absorbed but rather sent in a different direction"? In the low energy region h V « m c 2 , scattering cross-section aln be cal- culated classically and is given by the Thomson foxmSia: elec+ror^Vt^i^^^f 12 ^ ^"t^nagMtt* wave impinges on an AsluS The JtJ leCt r n / 3 SSt ln m ° ti0n and therefore radiates. ,^T J*f el ^ctron is free, a good approximation If ^ of the is tha ^rT h h^G^ than the frequency of the electron I? A is the acceleration of the electron, " e£ =rmA PLANE POLARIZED A^ A = PHOTOH K ■ar L FIG. 11,12 "" f ELECTION ELECTRIC VECTQK £ OF £, M. WAVE. Ch. II Compton Scattering *L The instantaneous power radiated from a moving electron is* dt " 3 c3 11.36 2 e P T 3 c 4 /w>* In this case the average rower ia j£ ar This energy taken from the incident wave and re- radiated may- be expressed in terms of a cross- section C% per electron defined so that _, (total energy radiated per Oj X (Intensity of Incident beam)= un it time by the electron) Q^cra 2 ) X I (erg-cm^-sec"- 1 ') = yy (erg-sec^ 1 ) 11.37 This definition is equivalent to saying that all the energy in- cident upon a surface Cfr i s absorbed and re?- radiated. The intensity of the incident beam is I --T^sz. (evtf-o^-sec ') Therefore U T — r _ lM = T' r e' = c - 66 x 10 "• -'(cm ) ( per electron ) II. 38 where r g Is the classical radius of the electron, 2 .8 x lO" 1 ^ em. The Thomson cross- section is evidently independent of fre- quency. 11.38 breaks down when h.p is near or above mc =^^Mev. Then the phenomenon must be treated as Compton scattering. In the Compton effect, an incident photon \-JJ collides with an electron. Compared with the energy of Irz* , the binding energy of the electron is negligible. The scattered photon suffers a change of energy which can be calculated by imposing conservation of relativistic energy and relativistic momentum on the collision**. The result is ■"Scattered — A incident = X^T \\ -Cos ©J 11-39 nV Incident Vixj sc&ti&red electron The frequency of the scattered photon is less than that of the primary* The factor h/mc«^ c is the Cometon wavelength, 0.024 x ID -8 cm. The cross-section for Compton scattering Is given by the Klein-Nishina formula***: 11.40 ^Abraham- Becker, Tbeorie der Electrizitfit , Vol. II. p. 73 **Thi's is done, for example, in Rlehtmeyer and Kennard, Intro - duction to Modern Physics, 3rd ed . p.. 533. *'«*Derived in Heitler, I.e., p. 149 42 ■ Compton Scattering Ch. II Problem: F ind asymptotic Bxnreualora for II 4n nZ i ^ enemies and for higJl energle5 _ ^ ^11 .40 'rouS^! " The asymptotic expressions are: .Low .energies: C = a T [ 1 -Jtw- + ^**+ ... j , <*«! H.4I HiSh energies: eg « fc^ka*-^) rt>>1 The Klein-Nishina formula is plotted below;* \AZ Direction of POLARIZATION rorJa n ^I?3 r 8 d L1Sf1L 0n a„4^ r / d T st en ?^i r where the Thomson 13 that due to a radiating c^atlica rtrll «* 1 « a ° f *»**tad ener^ ft'VS ^ ^ the an S le between iBBrit S 4 ■ - Primary radiation azW* P f 1 ^' an ai ™3* over the 5ffi£*}* ^* tne maxima of scattered CP J^p " hen ^f approaches and ex- tlAtA radiation is predominantly 'INCIDENT RADIATION FIG-. ¥+9^30° 11.13 lm,enSt e y S ifS foTow'nJ ■" ""S^ ai ***»M*» of scattered Jl a | +^(1-^.9) J H.43 Sw/J Tn?I * thS 1 J?S 1 ? f* Ate the direction 9^ ■ a cident radlatlonT^, Sein \i2^f ^ <* P laM polarized' 1?- from integrating this eqiaSon! ^ II,4 ° WOuId res ^ * From Janossy, p 05mlc Rays ( (0xford)i Ch. II Radiation by Fast Electrons 43 3 ■ Radiation loss by fast el ec trons {ore m s st rah lung ) . This paragraph should coin© under II " XJ sub-section 10, butls placed here be cause our calculation makes use of the Thomson formula for OV , derived in the last sub- section. Consider an electron of velocity V ^=c -passing a nucleus 2e, We calculate the energy given off by the electron in the forn of photons when it is accelerated by the field of the nucleus. FAST ELECTRON In the rest system of the electron, the nucleus Ze moves with velocity almost c. The electric field £ of the nucleus is con- tracted, and the associated magnetic field H is X. to £ and of about the same magnitude. Therefore the moving nucleus looks to the electron like a plane electromagnetic wave. This will be discussed more fully shortly. The wave of photons representing the nucleus suffers Compton scattering by the electron. These scattered photons, when viewed from the rest system of the nucleus (the laboratory system) appear as the photons emitted by the in- cident electron. According to this brief summary, the calcula- tion has the following parts: (a) Lorentz transform to the rest system of the electron. (b) Calculate- the characteristics of the Dulse of electro- magnetic radiation that represents the moving nucleus. (c) Find the density of photons corresconding to the pulse of electromagnetic radiation. (d) Calculate the probability of scattering of these photons by the electron. (e) Transform back to the laboratory frame. Throughout this calculation we leave out numerical factors. Their inclusion would obscure the Ideas involved. The correct numerical factor is Inserted at the end. (a) Let p = %«1 I i* IA/I-/3*' is large the electron in the laboratory frame is Ymc^. The energy of LAB, FRAME' ELECTRON FRAME Je FIG. II.15 44 Radiation "by Fast Electrons Ch. II In the lac frame, denoted by °, the electric field of the nucleus is \£°| = %*/&. In the electron frame, denoted *, Its perpendicular component £* is Increased by the factor H\ The magnetic f ield °&* seen by the electron in its own rest system i s _L to both £* and the velocity vector. Denote this field by ^ = | -A*\ . Then* Kj = € L %0 *^S ^ *| . V , Therefore y* = e * ( as ln a plaa@ electromagnetic wave. (b) The reel on in which the electromagnetic field is not zero has a length, In the electron frame, of about b/v , the l actor g coming from the Lorentz contraction SHAPF OFPUL5E: SPECTRAL DISTRIBUTION; t-0 when the electron is in wave- front Diane representing ■nucleus " ° FIG. 11.16 The spectral analysis of this pulse raay he done by assuming for simplicity that it is a Gauss! sn of time width b/fc,; e, -x, cc e 2be Z1 . 44 SSto^iSSli^? 11011 ° f C °^° nent ****»*« l- -then di ^Lsrtr.WiL^^ gst^Tr p 2 e ^^sr ic volume is dven bir S +X ari p- 3si I ?cY r _, n rrn ™p f^f x Y? n D/ Tjr;" p" V*F*7 ' , . Tae volume per unit area of the disc ecuals the thickness = b/y (cm Therefore the energy oer unit area Is ° ' re 1 ' ne (c) Approximate to the gaussian spectral distribution by a rectangle of width Xc/b ***■• This means that the energy carried by photons In the frequency range Ay* is simolv prooortkmal to *5h raham-Eecker, Vol. II, p. 48. " ~~ ~ **Strattoh, Electromagnetic Theory , p. 290. »**it will turn out later, (just before 11.54) that only frequen- cies in the electron system less than p* = mc^/h will be inte- grated over, and that tyj* » S**g .<< '|^fc , Therefore the portion of the rectangle used will be at a relatively flat oart of the par s si an , a s sho wn i n F I g . 1 1 . 1 7 . Ch - II Radiation by Fast Electrons <-o V. MSI. ^C therefore the energy carnled by photons d: this rr n - :e a*,* ^ m a at distance b, i§ P Az--*b ? 2 e^V The number of photons tier ~t 2 in the range A^ is equal to this XCenergj per photon^ 1 which is^)" 1 Therefore the number of oho tons £X + m l;V h ? ^anoncy" range T.r_oe bis giver, bv ; A^*at the dial FIG- 11.17 S&hS^'b b 3 ^c hv" o V y* > & b fj. i ™ f W tor ^ > * — I aj The mean number of o,„ t , Q „, ^ D frequency ^* occurri fi|i? the S SKTl" *** ?hotons of GQ X (density (per cnT 2 ) o Therefore No. of scattering everts photons of free. V* ) for rre a< ft~^ ^Wi^L *V A£* , « 01 h, am for one nucleus / ^h b a ^* c ^ J II .45 offfl^!S cross-section per nucleus be £fe*)*ft/*- ■ * whi^ Beyond b max ^X/^* ths f T&QV[miir y ,* (Y which we are commit 1 -ir- vr TJ *\ \ * ■. ^-^^.„/ ^ (for is the case and use max > W' - e assume this u min 'max is subject to seve: limitations. II .47 Sends on the d-.-m-i n,,-i~~-.,~ "~ """"'" Ul " L,J - u:!B ' Which is dominant de- must bel Lt " ^ The ni0st ^Portant is that b , ing the ^«1^S^;^^ n ^5«±«»« ^ Packet' reor^e^t- the Compton wavelength i * wl le ? S , h for the electron is velocity c/^ ,ithre S ^t to +X ^ *** *** electron have 4e Broglle vlvelength £ *° d *??,?** S ^ m > 1*1 a giving it velocities. Taking K^_ - f >-- 1 "mm - A Prna », tarn,-. a ',et: 4 = 137H"' 11,43 tm s pomt is w«asxs sjfgs/itif^^: 1 ^^^^-. m More nreciso Radiation by Fast Electrons acuiation leads to 133 Z' for II. 45. ^v*)av*^^*<1 c JU 173 §1? 5,0 SVSt« Fro: ( e ) T r a n s f o m. ":ac]i to the lab the photon changes from ~p* to ~i/ a Therefore Introducing this result into 11.49, '-Kh=-nr^e Ch. II Therefore II. ho The frequency of special relativity* 11.50 11.51 et; 11.52 The factor 4 /%. results from exact derivation. The cross- section per nucleus for scattering at frequency J/ in the lab system is ^•JA^Z^/A-H 11.53 where e~/c)i, the fine- structure constant, has been renlaced by 1/137 ■ Before integrating to find the total energy loss by the e 1 e c t ron , we e xam Ine the limits of validity of thi s re suit , In 11.49 the maximum value for Oc was used. This breaks down at [IV* = mc 2 , This defines the limiting value of I/* , namely Z^" . Corresponding to 7-i* are two values of ~U° , one for forward and one for backward scattering in the electron frame. (Thomson sc at t e r I ng 1 s symme t r i c al , f on;?, rd and ba c k . } The 1 a rge r value of ~P a results for a numerator in 11.50 of about two , i.e., Ke, = H * l+/3cr>vB* m Therefore the use of the Thomson scattering formula 11.38 causes the theory to break down wher- h^°/V = mc'-, or But thi s sav s 1/ :i!;.X enenrv of ele ctron 11.54 , which clearly must be so , because the electron cannot radiate a photon of energy great- er than its Initial energy. The expression 11.53 diverges at low frequencies. But the enerrv carried by low frequency photons is finite. The total TJ' per oath lenr.tb dx is energy loss to photon^ of freriuenc;. equal to : (energy ^>er |oinoton) X (no. of nucleytm^X (crass - section "£{V)AV") X &X Therefore -(cIeJrao. , the total average energy loss per path length dx for all frequencies Is given by: ft 4 ". JzjVsO 11.55 * cf. Abraham-Eecker, Vol. II, u, 312. Ch. II Pair Format i 4.7 ^-^^f^ II '.■.'".lore ! nzC = energy of s ].e< tron, ete^fT^t/lf^ 71 ^ '~ Xve an a FP re ^bl e amount of its rro^^e^oraSo^ffif^'l^? ^ 1 ***** l0M ** *** Wid ^ (fiE7dS) rad is evidently proportional to energy. It Is the re - hi 3 encr^ief *" C ° SmlC ra7 P h 7 3ics w ^e particles have very We may v/rite 11,56 in tenai Define Then X- -m-^^Y* (3e) = _ e ^c x -vat, F- s RID or °- ^ , the ra diation length , 11.57 jar For 3f«.4 , I/2.7- c -£ Roughly , rn E^E^e -" 11.58 the energy Is dissipated, on tho average whs re C is nearly constant. 11.59 Jk «C Radiation loss r>er g-civT 2 traversed is greater for heavy elements. The ratio of radiative energy loss to Ionization loss, for electrons, is given rouahlv by* ^Ens Material 1 n lEnergy for . ccsjiton ~ djxjiot-i. otX / RAD. , Air j NTP J Alurni:iu:i Lead "330 rn: 9,7 cm, .517 cm. ISO Mev 5S 7.0 ? * ( Me vj 800 H.6O (Thickness of the atmos- phere is equivalent to 8 fen. of WTP air, or about SOJJt ) + jor , Js e r^ rotation . We return to the discussion of absorp- tion 01 electromagnetic radiation in matter. The mos+ limoS S^i^S Ve T 7 r- lGh ener ^ ^ectroma^tir^atWirby SSnKSi ?" tMS PrCGBSS a ph ° ton diaappeara and a ros- Dir°c's ^;\ n 4ifi r ° n appear * This is t0 be underatood only by is slrr^ii^L ? 1C ™y e /«^hanics. The following discussion is aimplliied and qualitative**. According to the relatlyJ^Mctheory of the election an electron tea energy + xfu^WTZX^ mJ. \. Le ^~ -' ^ n lj -yinifi ) + p c . This equation permits "This appears .inverted, in Lecture XIII of Los Alamos Reoort #24 d\lt ?£ prin ~ Gd version. Also inverted in page 72 of fia'setti. Trie theory is given, for examole, in Heitler, I c 48 Pair Formation Oh. II I negative energy values. The energy spectrum of a free electron looks like Fig. 11,16. In Dirac' s theory, practically all negative states are filled at all points in space. A vacuum is then a sea of electrons in negative energy states. The presence of this charge is rot observed +<mc : because it is uniformly distributed. A photon of sufficiently high energy ■- I i FIG. II 18 may lift an electron from a negative energy state to a positive energy state The energy threshold for the photon is 2mc 2 , since for a free electron there are no states between -mc 2 and +nic 2 . Physically, this means that the photon must supply enough energy to create two particles of mass m. Momentum must be conserved and this requires either that the negative energy electron be near a nucleus or an electron, i.e., not free, or that two ohotons coming from different directions coalesce and lift an electron from a negative energy state*. If the electron is near a nucleus it may occupy discrete states just below +mc 2 . These are within a few e.v. of 510,000 e.v. Strictly, then, the threshold for pair formation near a nucleus is 2mc 2 - (binding energy of electron). This is of no importance because binding energy «mc 2 and because transitions from negative energy states to the dis- crete part of the spectrum are improbable and not yet observed. The result for cross- section for uair oroduction near a nucleus Is calculated in Heitler** and is II.61 In the extreme relativistic rai en rmr is independent of energy, For other energies the situation is more complicated, and is described in Heitler. (A simplified procedure similar to that used to comnute the radiation loss by fast electrons may be used to comoute the cross- section for pair formation at very high energies, if the orocess of collision of photons in a vacuum {h v + hv-*e + + e~J is known This is outlined. Consider a uhoton of \iv ^ 1000 mc 2 or so. Transform to a new coordinate system in which the nucleus moves very fast, almost, but not quite, c. The ohoton seen in the new system has reduced frequency. The nucleus looks like a wave of photons '(just as in subsection 3). Oive the nucleus such a vel- ocity that its photons have the same (reduced) frequency as the incident photon seen in the new coordinate system. The" process i ■:-■, the simpler one of collision of ohoton s in a vacuum, ' h v W- + e" ) shown in Appendix II. 4 that momentum cannot be conserved _r f OTVPra.t, 4 r nn Vnr r->no 1 onl a+ aA v-o, ,-,+ ,-. .« Xt is onunu j_n .tiuufciiJCLx j.1. ^f t,nat. momenta in pair formation by one isolated ohoton, *#Heitl g;r, i.e., p. 200 Ch.. II Pair Formation We may define a mean free path for pair production 49 Jp^ I II .61 Then the decrease in intensityynof a beam of photons will follow the equations: 7C dm b- „ £* /jt =^ Q e ^ 11.63 /n ^ 4- i 23 -Z a N _2 S 183 5 T37 yCe ^-^ 11.64 This is analogous to the equations for radiation loss by an electron, 11.58 and 11.57 e -i (H.58) E J* & Evidently 4m z H„a j U3. -*-•£-< (11.57) 11.65 COSMIC KAY 5. The phenomena of radiation and pair formation are responsible for the shower phenomenon in cosmic rays*. The shower is initiated by an electron haying energy of the order of billions of c.v. By radiation it produces photons of comparable energy. These photons produce electron-positron pairs, each particle of which radiates. The energy is soon divided among many particles. When the energy of a particle reaches the point where ionization loss predominates, the. shower stons. Me sons have ma s s about The factor (e 2 /mc2)2 appears formula for radiation loss, ^ meson, this factor is(l/2.Q0r 00 m. in the For a as large as for electrons, hence energy loss by. the meson by this process is much smaller. A meson looses energy mainly a meson of some billions of e.v. ionization the meson can penetrate several meters of 1 FIG. 11.19 by ionization. loss is small, ead . For and b. Summary . The phenomena of absorption of photons can be summarized in the following graphs. * The theory of showers is given in an article by Rossi and Greisen, Rev . Mod . Phys . 13_, 249 (1941 ) ' D ° Absorption of Photons GENERAL FEATURES OF ABSORPTION AS A FUNCTION OF ENERGY: Ch. II PHOTOELECTRIC ABSOF^PTIOW PAIR FORMATION _ .01 COMPTON EFFECT f (KLEIN -N1SMINA) ' l0 ENERGY^nric' 1 UHTO 100 FIG. II. 20 ABSORPTION COEFFICIENT FQ* PHOTONS iN LEAD, AS A FUNCTION OF ENE^T: * h z Ur U- U O o z 5 Q. B 8 8 \ Kw - limits nrrLC 1 FlS.II. 2.1 MATERIAL Alur Colpf3ey Lead Location of tninimum in total obsor^tion -35 MEV ~IO 3M* Thickness F j in 5/01' 1 f-w l f&f*$Q dedu ct mn an tntemi-iy ■ 47 &£m 3 at -35MW •31 -10 Mei -15 -3J£m< Problem: Suppose e ^gos-tUg ray electron of energy ld00~lFc 2 ~ tut the electron lec.ves the plate ^conroanled by a nair whose total energy (electron plus positron) is larger than^OO m^ 7 assume tne pr.rtic_Les chan-e direction nenlifri t>ly . Use 11.53, wri+lnc It in terms Q f^ ■ grf'^ ! To ? ;et the cross- section for all the nuclei ^thickness dx, by U dx, -ettlnr multiol: feU ^ * From Reltlt 216, Ch. II Aftiendix H.l cro 53 S9 ; u ^Sj hotons ir the ^ 5o ° - 2 *> 1000 BQ * J WjT - A ^°_ .693 j ~^77 - — ^— OA" Will This is the nrobabillti tha j - t. wu j be produced In LJ tSkSX dx^ " ° f ener * r > * The probability that a photon ,^ duced at X will then form a ,air fron 11.63. The pair Produced Will nave all the ener 3 of the that both- events will occur Is [using H.65 For lead in this energy region [^^ Ihereiore the probability is O.OO75 the scattering anrl e fn-n - =■ i " "■■ " > Where &. j S over .any traveraSs 0? the ^ftrbe^Tt^ ^ ^ lc ***«> alated angle between the IniUal df^.ff^ ^ b9 tKe net a ^uin- particle's velocity vector \^ ?£ Q ]J?^™* ^d the the traversal in ah&<?ti <-m a lii + re , " L1 . e -^.+ i collision o^ next paR e. W * atwm ' A Sorter derivation appears on the NOT IN PLANE OF PAPER IF i. ^o TRIANGLE 5 4+1 JM COLLISION-) A r -'V= ,to -^ + - l I^TIAL DirecT.ONTOWH.CH © k IS MEASURED FIC-. 11.22 This is a recursion^^I 2 ;, L d :^—^ *** respect to f, (Cos© p )=rTTc 05 « II. 67 ii™ S^Jiri, 8 * P^ 1 ' '"'- "^- «- -"•■ '" #•- #(.-£?« ST £ ? S j~ P**4 gj»- a- jf |s 52 Appendix II .3 Ch. II Shorter derivaxion of 11.22: ,„„+„,.,, We make use of the fact that small, angles are vector a. p Then Op = ?^. -i S£f 1=x $ =£*? *£*•** In averaging over many ■t riu s &*£% many traversals, Oj is positive as e_ — r 1~ often aa it is negative, Aonendix II .2 . Deri i greater detail. ?, the avsra C e number of collisions in one traversal is given fc y If f is a function o7 X M& N# is the normalized probability den sit;; forX , then the sean of f is J"=» fj%j ^fcrj d>K II . 59 anpl ice. 1 c a I cul at e ©\ S 1 = J [©0=) ] * ttfo) d b . The probability density for b, #J Is proportional %a prow.- sectional area at the distance b, 2Trb db , to the nuclear density N," and to the path length D. Therefore the normalized Wfe) I a Z ir lo D N 2 tt b DM Therefore Finally — i, f[&Cb)] 2 2-rrbDN4b II. '0 II. 71 ©*= ZTT (II Anoendix II.? . Equation 11.26 can be derived by consi dering the ^ ro lections "of the angles A on some chosen plane and deriving a value forV , the mean square of the resultant P™ -ted angle Then the unpro jected, true scattering an G ie can be found from MM result that ± @* = -q> a (^..<<,1) II * The factor £ comes fror. avera G in G cos 2 , the an G le bein G random, A "similar relation holds between the actual linear deviation ol Ch. II Appendix II ,3 53 Placement on some chosen pSmT'i!S! ^^ct-ion of that dis- /. emergence — f^— — I SlWpsy of entrance ^fc -g7^ 2 = -x 2 11.73 (plane of paper normal to Incident direction. Lookin 5 toward source 01 particles) In a cloud chamber Efe-oissrrartb 1+ i- a +1 scat terms that 'la shownf " ls the ? ro J G ^ed angle of -^^=^^ used bj danossy Problem.. Electrons of ensr-py 10° e.v. (from cosmic rays) pass through a lead nlate 1 G « thick Calculate the mean square , 01 the linear deviation of the electron emergent points from the peo- "" metrical shadow point on the- last the patii is almost straight. |^%«^^iS^^^S?S• ,, f ^-the^sur- The projected angle of deflexion at ^ n aZ ^ h tn Sle i S £ Projection of linear dissent £ 1" "f**^ - *» ^ DEVIATION » last irom the/^sur- for colli - sion =£; : PROJECTION \^" OP PAT h v ,. , *■ -D 1= & * FNCE TOWARD SOURCE OP Particles. **^pHj%%^ S 1§^&, bat C^c^ = X5 ^^ff / 3,Mce f* a "d i a r e independent " ^ "^ 6 2 P P= averse ho. of c.o|[ hS ions Th^un.^ected linear (dlspW^ht^ ^ , will be ^0. ^ 2 =^D*©* (^0.02 Cffl. USln£ H.31) 54 Arrendix II .4 3h. II g4& 11.74 A-Q^endix II .4 . Proof that moimnixw cannot be- conserved in hrz tion of an electron- positron pair by a sir.yle Isolated photon. By conservati on of energy : L By oonseri r ation of momentum: (The equality holds if 3 + and 3~ f/b ° r1 ' ln the same direction) Fro, n.75 (M*^ #c**4L*/e» + MM?^ Fro, II .74 M" = *fcV * * ^Vzt^lM^ Pf^^ x 2 The radical terra is larper than 2yjo +/ f5./o 2 , so both carrot be satisfied . References: Books and articles referred to ;:ore than once in this chapter are listed. Ab raham- Be c k e r , The orie der Slectrlzitat , Vol, II ( T eub re r , Ed.uards) Heitlcr, Quantum Theory of Radiation , 2nd ed . (Oxford) Janossy, Cosmic Ray s (Oxford) Livingston and Be the , Rev . Mod . Phy s . , £, 245 (1937) (experimental .nuclear phy s i c s ) Rutherford, Chad wick and Ellis, Rad i at i c n s f ro n Radi oa c t iv e Substances , ( C a mb rid ge~) Rossi and Greisen, Rev . Mod . Phys . 13, 249 (1941)